Diferensiyel operatörlerin düzenli izleri ve spektral özellikleri

This thesis consists of five main chapters. In introduction, we give a general information about the theory of Sturm-liouville operators and previous works in the literature which is realatively close to our studies. Also establishment of the problems is given in introduction. In the second chapter, we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with discontinuous weight at two interior points together with spectral parameter-dependent boundary conditions. By modifying some techniques of [C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293-308; O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulas for the eigenvalues and eigenfunctions. In the third chapter, we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. Namely we consider the discontinuous eigenvalue problem which consist of Sturm-Liouville equation on together with eigenparameter-dependent boundary conditions and transmission conditions at the points of discontinuity where is a given real-valued function continuous in and has finite limits ; is a complex eigenvalue parameter; are real numbers; and This section organised as follows: Firstly we give operator formulation of the problem in a suitable Hilbert space (i.e., A self-adjoint linear operator is defined in a suitable Hilbert space such that the eigenvalues of the considered problem coincide with those of ) and then asymptotic approximate formulas of characteristic function derived for four distinct cases. Asymptotic formulas for eigenvalues and eigenfunctions of the problem is given and finally we show that the eigenfunctions of are complete in . In the fourth chapter, assuming is a separable Hilbert space, we consider the operators and generated by the differential expressions and respectively, in the Hilbert space , with the same boundary conditions where is a positive definite self-adjoint operator in and satisfies some additional conditions. Let the eigenvalues of the operators and be and respectively. In this section, firstly we investigate the spectrum and resolvent of the operators and . Finally, under the conditions (1)-(3) the following formula has been found for the regularized trace of : In the fifth chapter, we investigate the resolvent operator and completeness of eigenfunctions of a Sturm-Liouville problem with discontinuities at two points. The problem contains an eigenparameter in the one of boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space and suitable self-adjoint linear operator in it.